Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(26,28)(27,33)(29,37)(38,40);(1,2,6,5)(3,8,7,4)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,3)(2,4)(5,8)(6,7)(9,10,13,16)(11,14,19,17)(12,15,21,20)(18,22,24,23)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,2,6,5)(3,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4,6,8)(2,7,5,3)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,5,6,2)(3,8,7,4)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662387);